Even if you run out of time, you should still be able to complete the quest, you just won't get the costume chest as a reward. Poke the "Active Tasks" tab on the bottom lefthand side of the screen, then scroll through the various quests to figure out your next step. I finished the "Need for Steed" quest about a week after the time had expired on it, so I can tell you for a fact that not all of the quests disappear after the timer runs out. It only seems to be the hobby oriented quests that will disappear when they time out, i.e. the hair boutique, puppy pal, etc.